Find the projection of vector A = (3, 4) onto vector B = (1, 2).
Practice Questions
1 question
Q1
Find the projection of vector A = (3, 4) onto vector B = (1, 2).
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Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).
Questions & Step-by-step Solutions
1 item
Q
Q: Find the projection of vector A = (3, 4) onto vector B = (1, 2).
Solution: Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).
Steps: 9
Step 1: Identify the vectors A and B. Here, A = (3, 4) and B = (1, 2).
Step 2: Calculate the dot product of A and B, denoted as A · B. This is done by multiplying the corresponding components of A and B and then adding them together: A · B = 3*1 + 4*2.
Step 3: Perform the multiplication: 3*1 = 3 and 4*2 = 8. Now add these results: 3 + 8 = 11. So, A · B = 11.
Step 4: Calculate the magnitude squared of vector B, denoted as |B|^2. This is done by squaring each component of B and then adding them together: |B|^2 = 1^2 + 2^2.
Step 5: Perform the squaring: 1^2 = 1 and 2^2 = 4. Now add these results: 1 + 4 = 5. So, |B|^2 = 5.
Step 6: Use the formula for the projection of A onto B: Projection of A onto B = (A · B) / |B|^2 * B.
Step 7: Substitute the values we found: Projection = (11 / 5) * B = (11 / 5) * (1, 2).
Step 8: Multiply the scalar (11/5) with each component of vector B: (11/5 * 1, 11/5 * 2) = (11/5, 22/5).
Step 9: The final result is the projection of vector A onto vector B, which is (11/5, 22/5).
